About Intervals and integers.

A friend of mine and I were having a discussion about whether or not Intervals comprised only of Integers could be classified as an Interval in a mathematical sense. My friend was and still is on the opinion that the definition in our textbook (same definition as on Wikipedia) excludes Intervals in any other Universe from being actual Intervals. I on the other hand disagree. We agree on the statement that given an Interval [a;b] one would assume the Universe of real numbers. I on the other hand maintain that the following would also be an Interval albeit more restricted;
[a;b] on Z or
[a..b] or
etc.

To sum up: I say that Integer Intervals and Complex Intervals and so forth are types of Intervals. He maintains that Integer Intervals and Complex Intervals are not strictly Intervals by definition and are just called Intervals for coherency's sake.

Sorry for long post and this is my first post so if it's in the wrong forum I am sorry but, can anyone lend an opinion?